Monogamy Inequality for Any Local Quantum Resource and Entanglement

We derive a monogamy inequality for any local quantum resource and entanglement. It results from the fact that there is always a convex measure for a quantum resource, as shown here, and from the relation between entanglement and local entropy. One of its consequences is an entanglement monogamy different from that usually discussed. If the local resource is nonuniformity or coherence, it is satisfied by familiar resource and entanglement measures. The ensuing upper bound for the local coherence, determined by the entanglement, is independent of the basis used to define the coherence.

Figure 1

Relative entropy of coherence for the long-time local state $p|0⟩⟨0|+(1-p)|1⟩⟨1|$, discussed in the text, and the basis $\{\cos \theta |0⟩+\sin \theta |1⟩,\sin \theta |0⟩-\cos \theta |1⟩\}$, as a function of $p$, for four values of $\theta$. The coherence for $\theta =\pi /4$ is $\log 2-h(p)$, where $h(p)$ is the entanglement of formation of the global state. It is the bound given by Eq. (7).